General physics a1 week 4 work – mechanical energy

Vietnam National University Ho Chi Minh City
Ho Chi Minh City University of Technology

General Physics A1
Week 4: Work – Mechanical Energy

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Contents
❑ Scalar Product
❑ Work
❑ Kinetic Energy and the Work-Energy Theorem
❑ Power
❑ Gravitational Potential Energy
❑ Elastic (Spring) Potential Energy
❑ Conservative and Nonconservative Forces
❑ Conservation of Energy

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Scalar Product of Two Vectors
 The

scalar product of two vectors is written as



It is also called the dot product



θ is the angle between A and B

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Scalar Product is a Scalar
 Not

a vector
 May be positive, negative, or zero

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Scalar Product: An
Example


❑ The vectors: A  2iˆ  3 ˆj and B  iˆ  2 ˆj
 
❑ Determine the scalar product: A  B  ?
 
A  B  Ax Bx  Ay B y  2  (-1)  3  2  -2  6  4

❑ Find the angle θ between these two vectors:
A  Ax2  Ay2  22  32  13
 
4

4
A B
cos  


AB
13 5
65
4
  cos1
 60.3
65

B  Bx2  By2  ( 1) 2  2 2  5

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Contents
❑ Scalar Product
❑ Work
❑ Kinetic Energy and the Work-Energy Theorem
❑ Power
❑ Gravitational Potential Energy
❑ Elastic (Spring) Potential Energy
❑ Conservative and Nonconservative Forces
❑ Conservation of Energy

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Definition of Work W


The work, W, done by a constant force on an object is
defined as the scalar (dot) product of the component of
the force along the direction of displacement and the
magnitude of the displacement







is the magnitude of the force
is the the object’s displacement
Φ is the angle between

and

SI Unit



N•m=J
J = ( kg • m / s2 ) • m

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Work: Positive or Negative


Work can be positive, negative, or zero. The sign
of the work depends on the direction of the force
relative to the displacement



Work
Work
Work
Work
Work






positive: if 0°<  < 90°
negative: if 90°<  <180°
zero: W = 0 if  = 90°
maximum if  = 0°
minimum if  = 180°

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Example: When Work is Zero
A man carries a bucket of water horizontally
at constant velocity.
 The force does no work on the bucket
 Displacement is horizontal
 Force is vertical
 cos 90°= 0


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Special Case: Constant Acceleration

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Contents
❑ Scalar Product
❑ Work
❑ Kinetic Energy and the Work-Energy Theorem
❑ Power
❑ Gravitational Potential Energy
❑ Elastic (Spring) Potential Energy
❑ Conservative and Nonconservative Forces
❑ Conservation of Energy

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Kinetic Energy
 For

an object m moving with
a speed of v

 Kinetic

Energy is energy
associated with the state of
motion of an object
 SI unit: joule (J)
1 joule = 1 J = 1 kg m2/s2

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Work-Energy Theorem
 When

work is done by a net force on an object
and the only change in the object is its speed,
the work done is equal to the change in the
object’s kinetic energy

Wtot



K 2 K1

K

Speed will increase if work is positive
Speed will decrease if work is negative

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Work with Varying Forces





On a graph of force as a function
of position, the total work done
by the force is represented by the
area under the curve between
the initial and the final position
Note there could be negative work!
Straight-line motion
W  Fax xa  Fbx xb  ......
x2

W   Fx dx
x1



Motion along a curve
P2

P2

P2

P1

P1

P1

W   F cos dl   F|| dl  

 
F  dl

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Work-Energy with Varying Forces


Work-energy theorem Wtot = K holds for varying
forces as well as for constant ones

dvx dvx dx
dv x


 vx
ax 
dt
dx dt
dx
x2

x2

x2

x1

x1

x1

Wtot   Fx dx   max dx  

dvx
dx
mvx
dx

v2

Wtot   mvx dvx
v1

1 2 1 2

Wtot  mv2  mv1  K
2
2

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Spring Force: a Varying Force
Involves the spring constant, k
 Hooke’s Law gives the force







Where is the force exerted on the spring in the
same direction of x
The force exerted by the spring is
k depends on how the spring is made of. Unit: N/m.

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Work Done on a Spring
 To

stretch a spring, we
must do work

 We apply equal and
opposite forces to the
ends of the spring and
gradually increase the
forces
 The work we must do to
stretch the spring from
x1 to x2
x2
x2
1
1
W   Fx dx   kxdx  kx22  kx12
x1
x1
2
2

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Contents
❑ Scalar Product
❑ Work
❑ Kinetic Energy and the Work-Energy Theorem
❑ Power
❑ Gravitational Potential Energy
❑ Elastic (Spring) Potential Energy
❑ Conservative and Nonconservative Forces
❑ Conservation of Energy

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Power
Work does not depend on time interval
 The rate at which energy is transferred is important
in the design and use of practical device
 The time rate of energy transfer is called power
 The average power is given by




when the method of energy transfer is work

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Instaneous Power
Power is the time rate of energy transfer. Power is
valid for any means of energy transfer
 Other expression




A more general definition of instantaneous power

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Units of Power
 The


SI unit of power is called the watt

1 watt = 1 joule / second = 1 kg . m2 / s3

A

unit of power in the US Customary system
is horsepower


1 hp = 550 ft . lb/s = 746 W

 Units

of power can also be used to express
units of work or energy


1 kWh = (1000 W)(3600 s) = 3.6 x106 J

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Contents

❑ Scalar Product
❑ Work
❑ Kinetic Energy and the Work-Energy Theorem
❑ Power
❑ Gravitational Potential Energy
❑ Elastic (Spring) Potential Energy
❑ Conservative and Nonconservative Forces
❑ Conservation of Energy

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Work Done by Gravity and
Gravitational Potential Energy
Wgrav
Wgrav

Fs

w y1 y2

U grav,1 U grav,2

mgy1 mgy2
U grav,2 U grav,1

U grav

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Potential Energy
Potential energy is associated with
the position of the object
 Gravitational Potential Energy is the
energy associated with the relative
position of an object in space near
the Earth’s surface
 Shared by both the object and Earth
 The gravitational potential energy








m is the mass of an object
g is the acceleration of gravity
y is the vertical position of the mass relative the surface
of the Earth
SI unit: joule (J)

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Reference Level
A

location where the gravitational potential
energy is zero must be chosen for each problem




The choice is arbitrary since the change in the
potential energy is the important quantity
Choose a convenient location for the zero
reference height
often the Earth’s surface
 may be some other point suggested by the problem




Once the position is chosen, it must remain fixed
for the entire problem

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